Geodesic
Upper bound for number of integral polynomials of four degree with given order of discriminants
Trudy Instituta matematiki, Tome 24 (2016) no. 2, pp. 14-19.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we propose a new method for estimating the number of top-integral polynomials with given discriminates. We show asymptotically accurate height assessment in the case of polynomials of fourth degree. At the same time we used the methods of the metric theory of Diophantine approximations of dependent variables. 
@article{TIMB_2016_24_2_a1,
     author = {V. I. Bernik and O. N. Kemesh},
     title = {Upper bound for number of integral polynomials of four degree with given order of discriminants},
     journal = {Trudy Instituta matematiki},
     pages = {14--19},
     publisher = {mathdoc},
     volume = {24},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2016_24_2_a1/}
}
TY  - JOUR
AU  - V. I. Bernik
AU  - O. N. Kemesh
TI  - Upper bound for number of integral polynomials of four degree with given order of discriminants
JO  - Trudy Instituta matematiki
PY  - 2016
SP  - 14
EP  - 19
VL  - 24
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TIMB_2016_24_2_a1/
LA  - ru
ID  - TIMB_2016_24_2_a1
ER  - 
%0 Journal Article
%A V. I. Bernik
%A O. N. Kemesh
%T Upper bound for number of integral polynomials of four degree with given order of discriminants
%J Trudy Instituta matematiki
%D 2016
%P 14-19
%V 24
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TIMB_2016_24_2_a1/
%G ru
%F TIMB_2016_24_2_a1
V. I. Bernik; O. N. Kemesh. Upper bound for number of integral polynomials of four degree with given order of discriminants. Trudy Instituta matematiki, Tome 24 (2016) no. 2, pp. 14-19. http://geodesic.mathdoc.fr/item/TIMB_2016_24_2_a1/

[1] Van Der Waerden B. L., Algebra, Springer-Verlag, Berlin–Heidelberg, 1971 | MR | Zbl

[2] Beresnevich V., Bernik V., Goetze F., “Integral polinomials with small discriminants and resultants”, Adv. Math., 298 (2016), 392–412 | DOI | MR

[3] Bugeaud Y., Approximation by algebraic numbers, Cambridge Tracts in Mathematics, 160, Cambridge University Press, Cambridge, 2004, 274 pp. | MR | Zbl

[4] Davenport H., “A note on binary cubic forms”, Mathematika, 8 (1961), 58–62 | DOI | MR | Zbl

[5] Volkmann B., “The real cubic case of Mahler's conjecture”, Mathematika, 8 (1961), 55–57 | DOI | MR | Zbl

[6] Sprindzuk V., Mahler's problem in the metric theory of numbers, Translations of Mathematical Monographs, 25, Amer. Math. Soc., Providence RI, 1969 | MR

[7] Mahler K., “Uber das Mass der Menge aller S-Zahlen”, Math. Ann., 106 (1932), 131–139 | DOI | MR | Zbl

[8] Bernik V., Gotze F., Kukso O., “Lower bounds for the number of integral polynomials with given order of discriminants”, Acta Arith., 133 (2008), 375–390 | DOI | MR | Zbl

[9] Gotze F., Kaliada D., Korolev M., “On the number of quadratic polynomials with bounded discriminants”, Mat. Zametki (to appear) , arXiv: (in Russian) 1308.2091

[10] Gotze F., Kaliada D., Kukso O., “The asymptotic number of integral cubic polynomials with bounded heights and discriminants”, Lith. Math. J., 54:2 (2014), 150–165 | DOI | MR | Zbl

[11] Amer. Math. Soc. Transl., 140 (1988), 15–44 | DOI | MR | Zbl | Zbl

[12] Beresnevich V., “On approximation of-real numbers by real algebraic numbers”, Acta Arith., 90:2 (1999), 97–112 | DOI | MR | Zbl

[13] Bernik V., “The exact order of approximating zero by integral polynomials”, Acta Arith., 53:1 (1989), 17–28 (in Russian) | DOI | MR | Zbl

[14] Bernik V., Budarina N., Goetze F., “Exact upper bounds for the number of the polynomials with given discriminants”, Lithuanian Mathematical Journal (to appear) | MR